Exploring Chaos: Theory and Experiment

An exciting new way of teaching chaos in dynamical systems to undergraduates, using a combination of text and computer experiments.. This book is an elementary introduction to the theory of dynamical systems and chaos. The principal aim is to explore the deep relationship among dynamical systems, chaos, and fractals, and to uncover structure even where order seems to be absent. The purpose in this text is to understand some of the phenomena which are common across diverse systems, and investigate the mechanisms which make them so. The approach will combine relatively simple mathematics with computer experiments using the program Chaos for Java which has been developed specifically for this purpose. }This book presents elements of the theory of chaos in dynamical systems in a framework of theoretical understanding coupled with numerical and graphical experimentation. The theory is developed using only elementary calculus and algebra, and includes dynamics of one-and two-dimensional maps, periodic orbits, stability and its quantification, chaotic behavior, and bifurcation theory of one-dimensional systems. There is an introduction to the theory of fractals, with an emphasis on the importance of scaling, and a concluding chapter on ordinary differential equations. The accompanying software, written in Java, is available online (see link below). The program enables students to carry out their own quantitative experiments on a variety of nonlinear systems, including the analysis of fixed points of compositions of maps, calculation of Fourier spectra and Lyapunov exponents, and box counting for two-dimensional maps. It also provides for visualizing orbits, final state and bifurcation diagrams, Fourier spectra and Lyapunov exponents, basins of attractions, three-dimensional orbits, Poincar sections, and return maps. }